Homogenization of interacting media

We consider energies modelling the interaction of two media parameterized by the same reference set, such as those used to study interactions of a thin film with a stiff substrate, hybrid laminates, or skeletal muscles. Analytically, these energies consist of a (possibly non-convex) functional of hyperelastic type and a second functional of the same type such as those used in variational theories of brittle fracture, paired by an interaction term governing the strength of the interaction depending on a small parameter. The overall behaviour is described by letting this parameter tend to zero, using the terminology of Gamma-convergence and exhibiting an effective energy. Such energy depends on a single parameter and is of hyperelastic type. The form of its energy function highlights a microscopic optimization between microfracture and oscillations, mixing homogenization and high-contrast effects. It is interesting to note that the period and orientation of the microstructures giving rise to homogenization depend on the overall limit strain, and are not due to geometrical features of the media.